Automated Atomic Force Microscope and the Operation Thereof

ABSTRACT

Improvements for rapidly calibrating and automatically operating a scanning probe microscope are disclosed. A central component of the SPM is the force transducer, typically a consumable cantilever element. By automatically calibrating transducer characteristics along with other instrumental parameters, scanning parameters can be rapidly and easily optimized, resulting in high-throughput, repeatable and accurate measurements. In contrast to dynamic optimization schemes, this can be accomplished before the surface is contacted, avoiding tip or sample damage from the beginning of the measurement process.

This application claims priority from provisional application No.61/995,847, filed Apr. 21, 2014, the entire contents of which areherewith incorporated by reference.

BACKGROUND OF THE INVENTION

For the sake of convenience, the current description focuses on systemsand techniques that may be realized in a particular embodiment ofcantilever-based instruments, the atomic force microscope (AFM).Cantilever-based instruments include such instruments as AFMs, scanningprobe microscopes, molecular force probe instruments (1D or 3D),high-resolution profilometers (including mechanical stylusprofilometers), surface modification instruments, chemical or biologicalsensing probes, and micro-actuated devices. The systems and techniquesdescribed herein may be realized in such other cantilever-basedinstruments as well as AFMs.

An AFM as shown in FIG. 1 is a device used to produce images of thetopography of the sample (and/or other sample characteristics) based oninformation obtained from scanning (e.g., rastering) a sharp tip on theend of a cantilever 1010 attached to a chip 1030 relative to the samplesurface. Topographical and/or other features of the sample are detectedby detecting changes in deflection and/or oscillation 1040characteristics of the cantilever (e.g., by detecting small changes indeflection, phase, frequency, etc., and using feedback 1060 to returnthe system to the reference state). By scanning the probe relative tothe sample surface 1090, a “map” of the sample topography or othersample characteristics may be obtained.

Changes in deflection or oscillation of the cantilever are typicallydetected by an optical lever arrangement whereby a light beam isdirected onto the back of the cantilever opposite the tip 1010. The beamreflected from the cantilever illuminates a spot on a position sensitivedetector (PSD 1020). As the deflection or oscillation of the cantileverchanges, the position of the reflected spot on the PSD changes, causinga change in the output from the PSD. In addition changes in thedeflection or oscillation of the cantilever are typically made totrigger a change in the vertical position of the cantilever baserelative to the surface of the sample (referred to herein as a change inthe Z position, where Z is generally orthogonal to the XY plane definedby the sample surface), in order to maintain the deflection oroscillation at a constant pre-set value 1050. This feedback 1060 istypically used to generate an AFM image 1110.

Actuators 1080 are commonly used in AFMs, for example to raster thecantilever or to change the position of the cantilever base relative tothe surface of the sample. The purpose of actuators is to providerelative movement between different components of the AFM; for example,between the probe 1010 and the sample 1040. For different purposes anddifferent results, it may be useful to actuate the sample, thecantilever or the tip of the cantilever, or some combination of theseelements. Sensors are also commonly used in AFMs. They are used todetect movement, position, or other attributes of various components ofthe AFM, including movement created by actuators.

For the purposes of the specification, unless otherwise indicated, theterm “actuator” 1080 refers to a broad array of devices that convertinput signals into physical motion, including piezo activated flexures,piezo tubes, piezo stacks, blocks, bimorphs, unimorphs, linear motors,electrostrictive actuators, electrostatic motors, capacitive motors,voice coil actuators and magnetostrictive actuators, and the term“sensor” or “position sensor” refers to a device that converts aphysical parameter such as displacement, velocity or acceleration intoone or more signals such as an electrical signal, including capacitivesensors, inductive sensors (including eddy current sensors),differential transformers (such as described in co-pending applicationsUS20020175677A1 and US20040075428A1, Linear Variable DifferentialTransformers for High Precision Position Measurements, andUS20040056653A1, Linear Variable Differential Transformer with DigitalElectronics, which are hereby incorporated by reference in theirentirety), variable reluctance, optical interferometry, opticaldeflection detectors (including those referred to above as a PSD andthose described in co-pending applications US20030209060A1 andUS20040079142A1, Apparatus and Method for Isolating and MeasuringMovement in Metrology Apparatus, which are hereby incorporated byreference in their entirety), strain gages, piezo sensors,magnetostrictive and electrostrictive sensors.

AFMs can be operated in a number of different sample characterizationmodes, including contact mode where the tip of the cantilever is inconstant contact with the surface of the sample, and AC modes where thetip makes no contact or only intermittent contact with the samplesurface.

In both the contact and ac 1080 sample characterization modes, theinteraction between the tip of the cantilever and the sample induces adiscernable effect on a cantilever-based operational parameter, such asthe cantilever deflection, the cantilever oscillation amplitude or thefrequency of the cantilever oscillation, the phase of the cantileveroscillation relative the signal driving the oscillation, all of whichare detectable by a sensor. AFMs use the resultant sensor-generatedsignal as a feedback control signal for the Z actuator to maintainconstant a designated cantilever-based operational parameter.

To get the best resolution measurements, one wants the tip of thecantilever to exert only a low force on the sample. In biology, forexample, one often deals with samples that are so soft that forces above10 pN can modify or damage the sample. This also holds true for highresolution measurements on hard samples such as inorganic crystals,since higher forces have the effect of pushing the tip into the sample,increasing the interaction area and thus lowering the resolution. For agiven deflection of the probe, the force increases with the springconstant (k) of the cantilever. When operating in air in AC modes wherethe tip makes only intermittent contact with the sample surface, springconstants below 30 N/m are desirable. For general operation in fluid,very small spring constants (less then about 1.0 N/m) are desirable.

At the same time it is often useful in biology to measure the stiffnessof a sample, to distinguish DNA from a salt crystal for example. Imagesof the topography of a sample do not tell us much about stiffness.

With contact AFM it has been common to measure the interaction forcesbetween the cantilever and the sample with Hooke's Law, a relationshipdescribing the behavior of springs, where the force exerted by thespring, F, is equal to a constant characterizing the spring, k, times achange in position of the spring, x: F=kx. In the case of AFMs, thespring is the cantilever, the constant is the spring constant of thecantilever, and the change in position is a change in the deflection ofthe cantilever as measured by the PSD.

Early spring constant calculations were based on order-of-magnitudeknowledge. One of the first attempts to produce a more accuratedetermination has come to be known as the Cleveland method, after JasonCleveland, then a graduate student at the University of California,Santa Barbara. Cleveland, J. P., et al., A nondestructive method fordetermining the spring constant of cantilevers for scanning forcemicroscopy, Rev. Scientific Instruments 64, 403, 1993. The Clevelandmethod estimates the spring constant by measuring the change in resonantfrequency of the cantilever after attaching tungsten spheres of knownmass to the end of the cantilever. Cleveland claims that this methodshould be applicable to most cantilevers used in calculating force.

A few years after publication of the Cleveland method, a simpler methodfor estimating the spring constant noninvasively was published which hasalso come to be known under the name of the lead author, John Sader.Sader, John E., et al., Calibration of rectangular atomic forcemicroscope cantilevers, Rev. Scientific Instruments 70, 3967, 1999. Thetheory starts with the well-known relationship between stiffness, mass,and resonance frequency (k=mω²) which provides an intuitive way tocalibrate the stiffness of a cantilever in air by simply measuring itswidth, length, height and resonance frequency. However, the large errorin the thickness of the cantilever, which may also not be uniform, leadsto a poor estimate of the cantilever mass. Whereas modeling the inertialloading of the cantilever (the cantilever mass) is inaccurate, theviscous loading of the cantilever can be reliably modeled in fluids suchas air. In ambient conditions, the viscous loading is dominated byhydrodynamic drag of the surrounding air. This hydrodynamic drag dependson the density and viscosity of air, and the plan-view geometry of thecantilever. Because the thickness of the cantilever plays no role in thehydrodynamic drag, the viscous loading can be very accurately modeledfor any given cantilever shape and used to calibrate cantilevers. Insummary, the Sader method provides a straightforward formula forestimating the spring constant of a rectangular cantilever whichrequires the resonant frequency and quality factor of the fundamentalmode of the cantilever, as well as its plan view dimensions. The methodalso assumes that the quality factor is equal to or greater than 1,which is typically true if the measurement is made in air.

The spring constant of the cantilever provides the information necessaryto calculate the force exerted by the cantilever. In an AFM system wherethe sample is moved relative to the cantilever, the additionallyrequired information is the change in the deflection signal of thecantilever (as measured by the PSD in volts), as a function of thedistance the actuator moves the sample (which is a conversion intodistance units of the voltage applied to the actuator). The estimate ofthis relationship is often referred to as the calibrated sensitivity ofthe optical lever of the AFM, or just optical lever sensitivity (“OLS”).

Where a relatively rigid sample was available, early estimates of thesensitivity of the optical lever of an AFM were typically made bypushing the tip of the cantilever into the sample and using the voltageresponse of the PSD (which is a measure of the change in the deflectionof the cantilever, but since it is in volts not a measure that isdirectly usable for our purposes) taken together with the distance theactuator moves the sample to estimate the factor that can be applied toconvert other voltage measurements of deflection into distance. Thismethod has some disadvantages: (1) where the sample is soft, as istypically the case with biological samples, the response to pushing thetip into the sample will be nonlinear and therefore not useful forestimating the voltage response of the PSD or the distance the actuatormoves the sample; (2) where the sample is a biological sample, it maycontaminate the tip when the tip is pushed into the sample and here tooyield a nonlinear response; and (3) it is altogether too easy to damagethe tip of a cantilever when pushing it into a sample.

An xy graph can be used to show typical sample displacement on thex-axis (the more the sample is raised the further out we are on thex-axis) and tip displacement on the y-axis (the more the tip is raisedthe further out on the y-axis) when the sample is raised toward the tipduring the process of making an estimate of the sensitivity of theoptical lever of an AFM. At point A on the graph the sample and the tipare sufficiently far apart that neither exerts a force on the other.When the sample is raised from point A to point B however the distancefrom the tip is sufficiently small that the attractive Van der Waalsforce is operative and the tip is pulled toward the surface. At point Cthe Van der Waals force has overcome the spring tension of thecantilever and the tip has dropped to the surface of the sample. If wecontinue to raise the sample further at this point, the tip will remainin contact with the rising sample following it upward along the pathdesignated by points C, E and F. If the sample is retracted from point Fthe tip will again follow—even beyond point C where the tip had droppedto the surface of the sample when the sample was being raised (theextent the tip follows the sample will depend in part on the presence ofcapillary forces, a normal condition except when the AFM is operating ina vacuum or in liquids). At some point however the tip will break freefrom the sample and return to the null position line. The point at whichthe tip breaks free is point G.

The distance on the y-axis from point G to point F is an estimate of thechange in the deflection of the cantilever when the tip of thecantilever is pushed into the sample, just as the distance on the x-axisbetween those points is an estimate of the distance the actuator movesthe sample. The relation between these estimates gives us the factorthat can be applied to convert other voltage measurements of deflectioninto distance. As noted above this factor is usually referred to asoptical lever sensitivity, or OLS. In order to distinguish it from asimilar factor to be discussed below we will also call it the dc OLS.

Recently a group working a Trinity College, Dublin and Asylum Researchin California, as well as John Sader, have developed a method whichprovides a straightforward formula for estimating OLS noninvasively. M.J. Higgins, et al., Noninvasive determination of optical leversensitivity in atomic force microscopy, Rev. Scientific Instruments 77,013701, 2006. The method requires the thermal noise spectrum of thecantilever to be measured (on an AFM) and the fundamental mode of thespectrum to be fitted to the power response function of the simpleharmonic oscillator. The equipartition theorem is then applied to thisresult and after simplification the result is a straightforward formulafor estimating inverse dc OLS which requires the spring constant of thecantilever, as well as its resonant frequency, quality factor and dcpower response.

We have discussed above the dc OLS factor estimated by raising andretracting the sample relative to the tip of the cantilever andcomparing the deflection of the cantilever as measured on the y-axiswith the distance the actuator moves the sample as measured on thex-axis. The dc OLS factor pertains to operation of an AFM in the contactmode. In ac modes a different OLS factor may be estimated. In ac modesthe cantilever is initially oscillating at its free amplitude unaffectedby the sample. In an AFM system where the sample is moved relative tothe cantilever, as the sample is raised to the oscillation path howeverthe tip of the oscillating cantilever contacts the sample and theamplitude of the vibration decreases. Initially for each nanometer thesample is raised the amplitude of the vibration decreases by a nanometeras well. When the sample begins to retract the amplitude increases untilthe tip is free of the sample surface and the amplitude of thecantilever oscillation levels off at its free amplitude. The ac OLSfactor is estimated by comparing the cantilever deflection amplitude asmeasured on the y-axis for one complete extension and retraction of thesample actuator with the distance the actuator moves the sample asmeasured on the x-axis. As with the dc OLS factor, cantilever deflectionamplitude is measured by the PSD in volts and the distance the actuatormoves the sample is a conversion into distance units of the voltageapplied to the actuator.

As with the spring constant and dc OLS, a noninvasive method formeasuring the ac OLS factor has also been developed. R. Proksch, et al.,Finite optical spot size and position corrections in thermal springconstant calibration, Nanotechnology 18, 1344, 2004. The authors makethe point, as had others before them, that the value of OLS will bedifferent depending on whether the cantilever is “deflected by alocalized and static force at the end [that is the end where the tip islocated], as in the case of force measurements or AFM imaging, orwhether it vibrates freely . . . .” They derive a factor for relatingthe two OLSs given by κ=InvOLS_(free)/InvOLS_(end). It will be notedthat the authors' invOLS_(free) measurement is essentially the ac OLSmeasurement discussed above and the InvOLS_(end) measurement isessentially the dc OLS measurement discussed above. In their discussionof the derivation of the factor for relating the two OLSs the authorsmake the point that the factor is dependent on the location of the laserspot on the cantilever.

SUMMARY OF THE INVENTION

The inventors recognize that a central component of the SPM is the forcetransducer, typically a consumable cantilever element. By automaticallycalibrating transducer characteristics along with other instrumentalparameters, scanning parameters can be rapidly and easily optimized,resulting in high-throughput, repeatable and accurate measurements. Incontrast to dynamic optimization schemes, this can be accomplishedbefore the surface is contacted, avoiding tip or sample damage from thebeginning of the measurement process.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1: Prior Art AFM.

FIG. 2: A plot of maximum integral gain versus sensitivity (InvOLS).

FIG. 3: A schematic of a preferred embodiment.

FIG. 4: Using the method to make optimized force curves and force maps

FIG. 5: A plot of a thermal spectrum (Amplitude vs. Frequency)demonstrating model fitting, Q, and spurious noise signals.

FIG. 6: Extending the method to operation in fluid

FIG. 7: The power law relationship between stiffness and resonancefrequency of the cantilever is shown for a batch of cantilevers of thesame model type.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Among the most significant challenges in setting up an AFM in ac mode isoptimizing the scanning parameters such that the tip of the cantileverand the sample are not damaged or fouled and the fidelity of the data isnot compromised. In particular the inventors have been concerned withintegral feedback gains and the scanning rate. When the integral gainsare set too low, the cantilever does a poor job of tracking the samplesurface and will cause damage or return a low fidelity image of thesurface. If they are set too high, the cantilever actuator can becomeunstable and oscillate which can cause severe tip and sample surfacedamage as well as returning a low fidelity image of the surface.

Some users skilled in the art start their setup of an AFM in ac modewith nominal parameters and adjust those parameters while scanning usingcommon sense. This method can result in damage to the tip of thecantilever or the sample because the starting parameters are notoptimal, and the parameters may only be optimized by the user afterdamage has be imparted on the cantilever tip. An alternative method,sometimes called the dynamic method, simply plunges in by engaging thetip on the sample and varying the feedback gains to “optimize” imagefidelity and hopefully reduce tip-sample damage. Whether this approachgives better results than the common sense approach is certainlyquestionable.

It is therefore desirable to have a reliable method for determiningoptimized imaging parameters for AFM in ac mode before the tip touchesthe sample. However this situation presents a particularly difficultproblem because the tip-sample interaction is highly nonlinear andtherefore not particularly amenable to simple analysis or prediction.Nevertheless we have extracted some simple relationships for optimizinga few imaging parameters. We begin with the optimization of integralgain.

Across many experiments we have derived a relatively simple empiricalrelationship between the ac optical lever sensitivity, “OLS” andintegral gain which allows an AFM user to set integral gain inaccordance with their tolerance for the risk of instability in thefeedback loop. FIG. 2 shows the relationship, a relationship between theinverse of ac OLS (which we call InvOLS) on the x-axis and integral gainon the y-axis. As can be seen, the line indicating the relationship hasa positive slope, indicating that as InvOLS increases so does integralgain.

The InvOLS/integral gain line was derived from two cantilevermeasurements on a sample of 25 different cantilevers of similardimensions. The measurements were made on the same ac mode AFM. Theinitial measurement on each cantilever was the determination of InvOLSwith integral gain fixed at a nominal amount. Immediately thereafter theAFM was engaged on a sample and the integral gain increased slowly untilthe cantilever actuator started to oscillate and became unstable. Aftercompletion of the sampling, the InvOLS variables taken at the beginningof each cantilever measurement were regressed on the integral gainvariables taken at the point of instability. The positively sloped lineFIG. 2 was the result.

The FIG. 2 InvOLS/integral gain line allows the user to choose the levelof risk of instability in the feedback loop they wish to accept. Afterloading a new cantilever in the AFM and measuring InvOLS with thatcantilever, the user can then select a high level of risk by settingintegral gain at the amount on the FIG. 2 InvOLS/integral gain linecorresponding to InvOLS as just measured. Of course the user couldinstead be more conservative and select a lower level of risk by settingintegral gain below the amount on the FIG. 2 InvOLS/integral gain linecorresponding to InvOLS, or even be more aggressive and select a riskyalternative by setting integral gain above the amount on the FIG. 2InvOLS/integral gain line corresponding to InvOLS.

To improve the fidelity of the imaging process, it is important toensure the cantilever is always sampling the topography. A cantileveroscillating at or near its resonance frequency requires some time toreact to changes in the tip-sample interaction, such as might take placewhile imaging a sample. These changes might be topographic or in thetip-sample stiffness for example. This relaxation time is proportionalto the quality factor, Q and inversely proportional to the resonancefrequency. Thus, low resonant frequency, and/or high Q factorcantilevers relax more slowly than do high resonance frequency, low Qfactor levers.

In either case, if a particular measurement is made too rapidly, beforethe cantilever can relax, a majority of the tip-sample interactions areeither made without the tip interacting strongly with the sample(parachuting) or with the lever interacting too strongly with thesample. Either case represents an error in the setpoint amplitude. Thiserror in turn may well lead to a mis-estimation in the tip-sampleinteractions and to other problems including poor, unstable topographicfeedback and poor estimation of surface dimensions and properties. Thus,it is desirable to operate the microscope in a manner that allows thecantilever to relax sufficiently to provide good feedback and reliablesurface measurements. One preferred means of accomplishing this is touse the formula below to place a maximum constraint on the scan rate SR,as the cantilever with resonant frequency f₀ and quality factor Qsamples a number of data points in a single scan line N is given by

${SR} \leq {\frac{\pi \; f_{0}}{2\; {Q \cdot N}}.}$

Thus for example, if we had a cantilever with Q=150, f₀=70,000 Hz(typical for a lever such as the Olympus AC240) scanning at N=512 pointsper line, the line scan rate should be

${SR} \leq \frac{{\pi \cdot 70},000\mspace{14mu} {Hz}}{2150 \cdot 512} \approx {1.4\mspace{14mu} {{Hz}.}}$

Those skilled in the art will recognize that as a reasonable upper limitfor the scan speed of that type of cantilever.

The measurement time for a pixel, a single point is

${SR} \leq {\frac{\pi \; f_{0}}{2\; {Q \cdot}}.}$

Similarly, the acquisition time for an image of N×M pixels isconstrained by

${SR} \leq {\frac{\pi \; f_{0}}{2\; {Q \cdot N \cdot M}}.}$

The adjustment of the imaging feedback parameters is reliant on knowingthe InvOLS of the cantilever. When various cantilevers are loaded intoand AFM, the deflection to voltage sensitivity that they exhibit—thevoltage is output by the detection means for a given deflectiondistance—varies from lever to lever. In the case of an optical lever,the sensitivity may vary due to variations in the positioning, focus,size and location of the spot. There may be variations in the smoothnessand uniformity of the portion of the cantilever involved in reflectingthe optical signal which will affect the sensitivity. Other detectionmethods such as piezo- or strain-resistive, interferometric and othersare also subject to variations from lever to lever.

As discussed previously, measurement of the InvOLS of the cantilever canpresent additional complexity while also risking damage or fouling ofthe tip or sample. Prior art methods exist for measuring the cantileversensitivity (InvOLS) without touching the surface of the sample. In apreferred method of accomplishing this, the thermal (Brownian) motion ofthe cantilever is measured. This information is then applied to ahydrodynamic function [Sader et al., REVIEW OF SCIENTIFIC INSTRUMENTS83, 103705 (2012), Green at al. Rev. Sci. Instrum., Vol. 75, No. 6, June2004, Sader et al. Rev. Sci. Instrum. 70, 3967 (1999)] to estimate boththe InvOLS and the spring constant. Note that this can be accomplishedon a single or on multiple resonance modes. This includes higherresonance modes that can be used for conventional imaging as well as avariety of other advanced techniques such as stiffness and modulusmapping.

When using a non-contact method to determine the InvOLS, the inventiondescribed here may use this parameter to optimize the scanningparameters such as feedback gains, cantilever amplitude, and samplingrate—without first touching a surface. Note that this is fundamentallydifferent from prior art approaches that dynamically adjust gains,amplitudes, and sampling parameters during the scanning process. Inthose approaches, if the initial parameters are chosen to be too high ortoo low, the tip and the sample are likely to be irreversibly damaged.

This novel non-contact parameter optimization method can optionally beused in conjunction with dynamic adjustments of some parameters afterthe initial engagement between the cantilever and the sample hasoccurred. If for example, the roughness of the sample turns out to begreater over one area than it is over another or than the expected inputparameter, the cantilever amplitude and setpoint could be adjusteddynamically to account for this change. Other metrics such as cantileverphase, feedback loop ringing, or differences in the trace and retracescans can be used to optimize the imaging parameters during the scan.These optimization criteria can be applied and adjusted on ascan-by-scan, line-by-line, or pixel-by-pixel basis. Even though thisimplies that the original settings were sub-optimal, the initialparameter settings afforded by this method can ensure that when thedynamic optimization is performed, it is accomplished with initialsettings that avoid damage or fouling to the tip or sample.

Q-control, preferably digital Q-control as that described in U.S. Pat.No. 8,042,383 can also be used in conjunction with the method. TheQ-gain can be adjusted to provide for faster operation of a cantilever(lower Q) or increased sensitivity (higher Q) depending on (i) the usergoals or dynamically based on the measured response of the cantilever tothe sample being measured.

There are many methods for measuring the spring constant and sensitivityof cantilevers. A recent review of some of these methods is contained inJ. E. Sader, Review of Scientific Instruments 83, 103705 (2012) and thereferences. Some of these methods, such as the so-called “thermalmethod” (J. L. Hutter and J. Bechhoefer, Rev. Sci. Instrum. 64, 1868(1993).) require that the sensitivity is also characterized. Thesemethods yield the spring constant as a function of the sensitivity. Ifthis is the case and if the spring constant of the cantilever is knownthrough a separate, independent method, then the sensitivity can beinferred by inverting the first spring constant sensitivity function. Ifboth the methods are non-contact, then the sensitivity can be estimatedwithout requiring that the tip touch the sample. This method ofsimultaneously estimating the spring constant and sensitivity has beencommercially implemented under the trade name “GetReal”. In this case,the “Sader method” described in the following paragraph is used toestimate the spring constant and then the “thermal method” is used toestimate the sensitivity.

Sader's original theory described, in the background of the inventionsection, models the hydrodynamic drag of the cantilever and itsfrequency dependence for a rectangular cantilever. Since then, Saderproposed a new theory that relies on empirically measuring the frequencydependence of the hydrodynamic drag function for any arbitrarycantilever shape [John E. Sader, Julian A. Sanelli, Brian D. Adamson,Jason P. Monty, Xingzhan Wei et al. “Spring constant calibration ofatomic force microscope cantilevers of arbitrary shape” REVIEW OFSCIENTIFIC INSTRUMENTS 83, 103705 (2012)]. Once the hydrodynamic dragfunction of a cantilever is known, it can be related to the mass of thecantilever by a careful measurement of the Q factor, which is equal tothe ratio of the inertial loading over the viscous loading. Therefore,the product of the empirically determined viscous loading (by the AFMmanufacturer) and the carefully measured Q factor (measured by theexperimenter) provides a precise estimate of the inertial mass loadingof the cantilever. With a known mass of the cantilever, the well-knownrelationship k=mω² can be applied to calculate the stiffness.

A preferred implementation of the invention is described in FIG. 3. Inthis case, the AFM of FIG. 1 is enhanced with analog or digitalcomputation means 4010. Non-contact measurements of the cantileverresponse 4020 to either Brownian motion or optionally driven 1100cantilever response is used, in conjunction with optional user definedgoals 4030 to estimate optimal imaging and/or measurement settings forthe AFM. For example, the setpoint 4040 can be automatically chosenbased on the user preferences and response of the cantilever. Inaddition, the cantilever excitation parameters, such as drive amplitude,frequency and phase can be controlled 4040. The gain(s) controllingvarious feedback loops can also be automatically optimized 4050. The xyscanning parameters 4060 can also be optimized dependent upon theuser-defined goals 4030 and the non-contact measurements 1020.

This method can be used to automatically configure an AFM to make forcecurves measurements. A force measurement is shown in FIG. 4. An actuator5010 that causes relative motion between the base of a cantilever 1030and a sample 5020 often results in deflection of the cantilever 1010 asthe tip of the cantilever interacts with the surface. A typical sequenceof events is illustrated by a plot of the cantilever deflection outputby the detector 1020 (in volts) versus the base position (typicallymeasured in convenient units of length, often nanometers ormicrometers). Initially, the cantilever approaches the surface withsmall deflection 5030, perhaps resulting from long-range forces betweenthe tip and the sample or the cantilever and the sample. When the tipencounters the strong short ranged repulsive forces, it begins todeflect more significantly 5040. At some point, the point of maximumforce 5050, the cantilever-sample motion reverses and the force betweenthe tip and sample begins to decrease. If there are adhesive forcespresent, there can be a snap-off 5060 and then the measured deflectionreturns to the initial baseline 5030. If the sample is hard enough tonot be significantly indented by the cantilever tip, then the verticaldeflection of the cantilever should match the motion of the sampledriven by the actuator. In this case, the sensitivity of the positionsensor can be estimated from the slope of the repulsive contact line5040. If the sample is compliant, then the slope of that line (or moreoften curve) provides information regarding sample mechanical propertiesincluding the modulus, plasticity, hardness and many other propertiesknown to one skilled in the art. In practice, since the tip can bedamaged with this sort of calibration and because most samples have atleast some unknown indentation, it is preferable to estimate the InvOLSin some other, preferably non-contact manner.

Non-contact calibration which utilizes the preferred method of measuringa Brownian or optionally driven cantilever as shown in FIG. 6 can bedifficult to automate. The amplitude versus frequency spectrum 6010 ofthe cantilever is plotted and the resonance frequency is found at anominal value 6020 based on the cantilever type. Hydrodynamic models arefitted to the resonance peak 6030 and used to calculate the springconstant and subsequently the InvOLS, as described above. Fitting theresonance peak to a hydrodynamic function can be difficult to automateas the measured data can contain multiple driven harmonics or spuriousnoise peaks 6040 from the environment or the electronics of theinstrument. Fitting these features would result in incorrect calibrationand failure to calculate the correct scanning parameters. In order toreduce parameter selection errors based on improper automated fitting,the current implementation utilizes the expected frequency range of thecantilever based on input either from the user or from some automatedmethod which identifies the cantilever being used and constrains thefitting range to that of the expected first fundamental. Further,applying strict selection criteria on the resultant measured Q based onthe width of the peak 6050 and white noise of the measurement that is tobe fitted can be used to reject noise peaks. Peaks that do not fallwithin the expected Q and white noise parameters of the chosen lever arerejected from the fitting routine. Finally, the user may override thisautomated implementation and manually fit the models to determine thesensitivity and spring constant of the cantilever.

In addition, since control of the maximum force involves gain parameters(see for example see The Jumping Probe Microscope in U.S. Pat. No.5,415,027 and its numerous family members as well and its derivativessuch as Pulsed Force Microscopy (A Rosa-Zeiser et al., Meas. Sci.Technol. 8 (1997) 1333-1338.), U.S. Pat. No. 8,650,660 and relatedfamily members, marketed as “Peak Force Tapping”), this provides asuperior solution to gain optimization than does iterative optimizationsince the iterated methods invariably involve several measurement pointsthat are non-optimized. Note that this does not preclude using thismethod and then further optimizing the imaging control parametersdynamically or iteratively. In some cases, it may be desirable to make amore conservative estimate of gain parameters for example and to thendynamically optimize the gains and/or other parameters.

The inventors have found that this method works excellently for bothtapping mode imaging, where the cantilever is driven at or near itsresonant frequency and jumping probe microscopy where the cantilever isdriven or oscillated at a frequency below resonance.

This method also provides a substantial improvement over combinedscanning methods. This includes two- or multi-pass methods that make useof stored data. An important example of height variation as a method ofdecoupling short and long range forces is the well-known double-pass orinterleave mode (“lift” or “nap”), first pioneered by Hosaka et al S.Hosaka, A. Kikukawa, Y. Honda, H. Koyanagi and S. Tanaka, Jpn. J. Appl.Phys., 31, L904-907 1992). This was also commercialized and described ina US patent. In these dual or multipass modes, the first AFM point, lineor full two-dimensional image scan is used to determine the position ofthe surface (measure the topography), i.e., the condition at which themeasured signal, R(h,V₀)=R₀, where R₀ is a set-point value. The feedbacksignal, R, can be static deflection for contact mode AFM, oscillationamplitude for an amplitude-based detection signal, or frequency shiftfor frequency-tracking methods. The second scan is performed todetermine interactions at a constant distance or bias condition tomeasure=R(h+δV₁).

As a typical non-contact force example, MFM and EFM measurements can bemade of the relatively long ranged and weak magnetic and/or electricforces. Here, once the position of the surface has been determined,force measurement at positive (=0-500 nm above the surface) separationyields magnetic (if probe is magnetized) or electrostatic (if probe isbiased) force components. Some versions of these non-contact forcemeasurement techniques may utilize a ‘negative’ separation, which occurswhen the surface height and the force measurement are made withdifferent cantilever oscillation amplitudes.

Short ranged elastic and viscoelastic forces can be measured byapproaching closer to the surface (negative δ, where δ=the separation ofheight between the surface and the force measurement). In this case, thetip will penetrate further into the sample, thereby probing short rangedrepulsive interactions. Similarly, by increasing the range of the tipexcursion, both short and long ranged elastic and viscous (dissipative)interactions can be probed.

A recent application of this method to force measurements is describedin Nanotechnology 22 (2011) 295704, the total article is hereby includedin this application. In this implementation, the second pass is used toprovide Jumping Probe or Force Curve-like information of the tip-sampleinteractions, using information about the sample topography measuredfrom the first tapping mode pass. This method is greatly improved byoptimizing the gain and scanning parameters as described here.

In some cases, there is more than one oscillation mode being used,especially for materials properties measurements. Examples are discussedat length in a U.S. Pat. No. 8,024,963 and family members by some of theinventors here. In some cases, there are additional feedback loopsassociated with operating for example, the second resonant mode of thecantilever in a phase-locked loop or other means of tracking theresonant frequency. There could also be gain control on the amplitude ofthe second mode used to keep it constant, improving the operation. Thegains required to operate the microscope using these modes have formerlyrequired user inputs and adjustments. By calibrating the stiffness andsensitivity of these modes, a curve similar to that shown in FIG. 2 canbe generated and then used to automatically set these gains. Thisprovides greatly improved ease of use and stability for these advancedimaging and measurement modes.

This method can be extended to operation in fluid environments. Onemethod consists of first characterizing the spring constant in air.Then, when the cantilever is put into a fluid environment, the InvOLScan be deduced from the assumption that the spring constant as measuredin the two environments is the same. By using the relationship

${k = \frac{k_{B}T}{{InvOLS}^{2}{\langle{\Delta \; V^{2}}\rangle}}},$

the InvOLS can be calculated as

${InvOLS} = {\sqrt{\frac{k_{B}T}{k{\langle{\Delta \; V^{2}}\rangle}}}.}$

This is illustrated in FIG. 6, where the amplitude spectral densities ofa single cantilever are shown in four different environments: in air farfrom a surface 7010, in air, close to a surface 7020, in water far froma surface 7030 and in water close to a surface 7040. In this particularcase, the InvOLS was measured by doing a force curve on the surface andso represents a test of the above method.

In some cases it may be preferable to omit the step of measuring thespring constant in air and to calibrate cantilevers directly in liquids.With the advent of automated AFM protocols that optimize imagingparameters based on predictive algorithms, direct calibration ofcantilever stiffness in different media has become a high priority.Pre-calibration of cantilevers in air is inconvenient for manyexperiments that require the injection of liquid as the first step oftheir protocol. Furthermore, calibrating a cantilever after it has madecontact with a liquid may be highly inaccurate because the damping maychange dramatically between air and liquid. Furthermore, contact withliquid causes irreversible changes to the damping properties to thecantilever once it is removed from the liquid due to surfacecontaminants that remain adhered to the cantilever even after it isdried. Therefore, it may be impossible to accurately calibrate acantilever in air noninvasively after it has been exposed to liquidusing Sader's calibration method.

In liquids, Sader's calibration methods break down because the Q factorof the cantilever is very difficult to measure accurately. In otherwords, the benefits of modeling the hydrodynamic viscous loadingaccurately are outweighed by the error in Q factor estimation. Howeverthe mass of the cantilever is overshadowed by the mass of the loadingfluid when the cantilever is immersed in a liquid. This is evident fromthe fact that the resonance frequency of the cantilever drops by afactor of ˜4× in water; i.e. the fluid mass is ˜16× larger than thecantilever mass. In this hydrodynamic regime, the effective mass of theoscillator becomes a function of the density and viscosity of the fluid,and the plan view geometry of the cantilever. Importantly, because thefluid mass loading is a hydrodynamic quantity, it is independent of thecantilever thickness and its uncertainty. So, rather than modelingviscous loading and measuring the Q factor, as was done for calibrationin air described previously, it is more accurate to simply model theinertial fluid mass loading directly when calibrating in liquids. Aftermodeling the fluid mass loading m_(f) and its frequency dependencem_(f)∝ω^(β), all that remains is a measurement of the resonancefrequency of the cantilever to calibrate the stiffness using themodified relationship k=m_(f)ω². This method avoids introducing Q factorerror into the stiffness estimation, as well as any error due to thecantilever thickness uncertainty.

To put this method into practice, it is necessary to empiricallydetermine the hydrodynamic fluid mass loading function for a givencantilever shape by measuring the resonance frequency and stiffness ofmany such cantilevers. This allows the determination of the β factor fora given cantilever type for a particular liquid, such as water. FIG. 7shows empirical data 8010 of cantilever stiffness versus resonancefrequency in water and the fit power law behavior 8020 for a specificcantilever model.

Once this hydrodynamic function is determined for a particular batch ofcantilevers in a particular liquid (by the AFM manufacturer), and theresonance frequency is measured in the same liquid (by theexperimenter), the stiffness of a single cantilever can be preciselycalculated. Importantly, the measurement of the resonance frequency isvery accurate down to Q factors as low as ˜1, making this method veryaccurate for calibrating most cantilevers in water, and many otherliquids.

In cases where the fluid mass loading m_(f) is comparable to thecantilever mass m_(c), it may be more accurate to model the frequencydependence of the total mass (m_(c)+m_(f)), where m_(c) is the nominalmass of the cantilever in air. Although m_(c) is prone to the errorscaused by the unknown thickness of the lever, these errors may benegligible as long as m_(c) is not larger than m_(f). As anapproximation, m_(c) may be assumed constant for all cantilevers.

For calibration in either air or liquids, at least one test cantilevermust be well calibrated using an independent method to obtain the truestiffness of the cantilever, and its corresponding resonance frequencyand Q factor for calibration in air or its corresponding resonancefrequency in liquid for calibration in that liquid. Such an independentcalibration may be performed with an interferometric detectiontechnique.

Since the optical sensitivity of an interferometer is determined by theinterferometer design and the associated wavelength of light, it issubstantially independent of the cantilever properties themselves. Inother words, for a given cantilever motion, the optical beam deflectionmethod will have a sensitivity (InvOLS) that will vary from lever tolever and system to system. Interferometers on the other hand, have asensitivity that is based on the well-defined wavelength of the lightused in the instrument. Because of this, many of the steps discussedabove can be omitted and the optimized gains of the system can bedetermined a priori without resorting to the steps outlined above.

The interferometer used to determine the sensitivities and otherproperties can be used independently or in conjunction with a differentcantilever detection method. An example of such an instrument isdisclosed in the co-pending patent application and in the paper A.Labuda and R. Proksch, accepted APL, attached by reference to thisapplication.

Preferred method:

With an atomic force microscope system operating to characterize asample:

-   -   1. measuring the sensitivity of the detector monitoring the        cantilever deflection;    -   2. Adjust the gain(s) of a feedback system that controls the        tip-sample separation based on an error signal and that is        controlled by at least one gain parameter;    -   3. Where we estimate one or more of the gain parameter(s) based        on the measured sensitivity.        In tapping mode and related techniques, the error signal is the        cantilever amplitude. Also, a common gain is the integral gain        parameter. It is also preferable to measure the sensitivity (and        additionally the spring constant without making contact between        the tip and the sample. This can be accomplished by    -   1. estimating the spring constant of the cantilever with one        method that depends on the detection sensitivity;    -   2. estimating the spring constant with a second method;        estimating the sensitivity of the cantilever detection by        inverting the first method using the spring constant estimation        from the second method.

Although only a few embodiments have been disclosed in detail above,other embodiments are possible and the inventors intend these to beencompassed within this specification. The specification describesspecific examples to accomplish a more general goal that may beaccomplished in another way. This disclosure is intended to beexemplary, and the claims are intended to cover any modification oralternative which might be predictable to a person having ordinary skillin the art. For example, other devices, and forms of modularity, can beused.

Also the inventors intend that only those claims which use the words“means for” are intended to be interpreted under 35 USC 112, sixthparagraph. Moreover, no limitations from the specification are intendedto be read into any claims, unless those limitations are expresslyincluded in the claims. The computers described herein may be any kindof computer, either general purpose, Or some specific purpose computersuch as a workstation. The computer may also be a handheld computer,such as a PDA, tablet, cellphone, or laptop.

The programs may be written in C, or Java, Python, Brew or any otherprogramming language. The programs may be resident on a storage medium,e.g., magnetic or optical, e.g. the computer hard drive, a removabledisk or media such as a memory stick or SD media, or other removablemedium. The programs may also be run over a network, for example, with aserver or other machine sending signals to the local machine, whichallows the local machine to carry out the operations described herein.

What is claimed is:
 1. A method of operating a cantilever basedmeasuring instrument, comprising: obtaining a relationship between anoptical lever sensitivity of a cantilever of the cantilever basedinstrument in a dynamic environment where the sensitivity depends ondistances to a sample, and using said relationship to determine adynamic optical lever sensitivity called invOLS of said cantilever; andmeasuring surfaces of the surface being measured using said invOLSvalue, by using a tip of the cantilever to measure characteristics ofthe surface and by estimating parameters of gain in the measurement,based on the invOLS value.
 2. The method as in claim 1, wherein saidobtaining said invOLS value comprises measuring a sensitivity of thecantilever of the cantilever based measuring instrument.
 3. The methodas in claim 2, wherein said obtaining said invOLS value comprisesmeasuring the sensitivity without making contact between a tip of thecantilever and the sample.
 4. The method as in claim 3, wherein saidobtaining a relationship comprises obtaining a frequency spectrum ofBrownian movement, and using said frequency spectrum to determine saidsensitivity.
 5. The method as in claim 1, further comprising determininga spring constant of the cantilever.
 6. The method as in claim 1,wherein said measuring comprises adjusting a gain of the feedback systemthat controls a separation between a tip and the sample based on anerror signal, that is based on at least one gain parameter that isestimated from the measured sensitivity.
 7. The method as in claim 3,wherein the sensitivity is measured by estimating a first springconstant of the cantilever using one technique that depends on detectionsensitivity and estimating a second spring constant with a secondtechnique different than the first technique, and estimating asensitivity of the cantilever detection by inverting the first springconstant against the second spring constant.
 8. The method as in claim1, wherein said cantilever based instrument is an Atomic ForceMicroscope.
 9. A cantilever based measuring instrument apparatus,comprising: a cantilever based instrument that has a cantilever,operating to measure a surface, a controller that controls measuringinformation about the cantilever, in a dynamic environment where thesensitivity depends on distances to a sample, and using saidrelationship to determine a dynamic optical lever sensitivity calledinvOLS of said cantilever; and said cantilever based instrumentoperating for measuring surfaces of the surface being measured usingsaid invOLS value, by using a tip of the cantilever to measurecharacteristics of the surface and by estimating parameters of gain inthe measurement, based on the invOLS value.
 10. The apparatus as inclaim 9, wherein said obtaining said invOLS value comprises measuring asensitivity of the cantilever of the cantilever based measuringinstrument.
 11. The apparatus as in claim 10, wherein said obtainingsaid invOLS value comprises measuring the sensitivity without makingcontact between a tip of the cantilever and the sample.
 12. Theapparatus as in claim 11, wherein said obtaining a relationshipcomprises obtaining a frequency spectrum of Brownian movement, and usingsaid frequency spectrum to determine said sensitivity.
 13. The apparatusas in claim 9, further comprising determining a spring constant of thecantilever.
 14. The apparatus as in claim 9, wherein said measuringcomprises adjusting a gain of the feedback system that controls aseparation between a tip and the sample based on an error signal, thatis based on at least one gain parameter that is estimated from themeasured sensitivity.
 15. The apparatus as in claim 3, wherein thesensitivity is measured by estimating a first spring constant of thecantilever using one technique that depends on detection sensitivity andestimating a second spring constant with a second technique differentthan the first technique, and estimating a sensitivity of the cantileverdetection by inverting the first spring constant against the secondspring constant.
 16. The apparatus as in claim 1, wherein saidcantilever based instrument is an Atomic Force Microscope.
 17. A methodof operating a cantilever based measuring instrument, comprising:obtaining a first relationship between an optical lever sensitivity of acantilever of the cantilever based instrument in a first environment todetermine a parameter of said cantilever in said first environment;obtaining a second relationship between an optical lever sensitivity ofa cantilever of the cantilever based instrument in a second environmentto determine a parameter of said cantilever in said second environment;and determining an optical lever sensitivity InvOLS for the cantilever,based on the spring constants in both environments being the same as${InvOLS} = {\sqrt{\frac{k_{B}T}{k{\langle{\Delta \; V^{2}}\rangle}}}.}$18. The method as in claim 17, wherein the first environment is in gas,and the second environment is in water.